Variational Problems: Nonsmooth Penalties and Time Scales

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Variational problems arise in a variety of disciplines, including engineering, natural resources, and economics. In the school of such optimization models, the class of Optimal Controls occupies an essential position since it stems from applications in a variety of fields. This class is distinguished by the presence of "hard" constraints such as the control and the state constraints. The generalized problem of Bolza, introduced by R.T. Rockafellar, includes under its wing the class of optimal controls. There, constraints are built into the objective function (or Lagrangian) through "nonsmooth penalty" terms. It is observed that better smoothness behavior is displayed by this model's Hamiltonian. The aim of the first part of the research is to complete the study of the generalized Bolza problem from the point of view of its Hamiltonian. The plan is to build a comprehensive second-order optimality theory and a stability analysis when the state constraints are present. The second goal of this research is to launch the study of optimal control problems over "time scales," that is, when the underlying time belongs to a compact set, not necessarily a connected interval. In particular, the aim is to develop necessary and sufficient criteria for optimality. Substantial modification of the latest techniques in continuous- and discrete-time optimal controls and in nonsmooth analysis will be instrumental in achieving the goals of this research.In recent years, systems in which the running time could be continuous or discrete have taken the front row due to their appearance in a wide range of applications. A time model that allows for such a combination and many more is known as a "time scale." This type of model is directly related to "hybrid systems," which occur in population dynamics, automotive electronics, automated systems, air traffic management systems, integrated system design, and multi-media. In these disciplines, one encounters applications that are formulated mathematically as continuous-time optimal control problems with constraints on the input and/or output, or as optimal control problems over time scales. For the former, important questions remain unanswered regarding finding accurate criteria to identify optimal candidates. On the other hand, since the latter is a completely new direction of research, very few results are known and several open questions need to be answered. The aim of this research is to develop criteria that serve in identifying optimal candidates for each of these classes of problems.